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New York • For eight intense, entertaining games, Sloane Stephens stayed right with Serena Williams.

For a 40-minute stretch in the fourth round of the U.S. Open, the top two American women put on quite a show. Both hit serves topping 115 mph. Both whipped groundstrokes to the corners. Both covered a lot of ground, extending points with leg-churning defense. Both showed the occasional sign of nerves, reflecting what a big deal this was, in part because the 15th-seeded Stephens already was one of only three players to beat No. 1 Williams this season.

Until, that is, the score was 4-all in the first set Sunday. That's when Williams took over.

The 20-year-old Stephens' time at the top of tennis may come. For now, the 31-year-old Williams is still as good as it gets. Taking eight of the last nine games, defending champion Williams returned to the quarterfinals at Flushing Meadows by beating Stephens 6-4, 6-1.

"When you give her that opportunity to take that step forward, she definitely makes her move," Stephens said. "Unfortunately, today she made her move. I just couldn't get back in."

Still, all in all, it was remarkably compelling and, within individual points, rather evenly played for what turned out to be such a runaway.

"I definitely think it was a high-quality match," said Williams, 64-4 with eight titles this year. "We both came out today to play."

She advanced to play No. 18 Carla Suarez Navarro of Spain, who defeated No. 8 Angelique Kerber of Germany 4-6, 6-3, 7-6 (3). Asked whether there's any chance of a letdown after getting past Stephens, Williams replied: "Absolutely not. I mean, I've been at this for a long time, so for me in my career, there are no letdowns."

In men's third-round action, the last of 15 Americans in the field lost, making this the first U.S. Open in history without at least one representative from the host country in the fourth round. Wild-card recipient Tim Smyczek, who is ranked 109th, was beaten 6-4, 4-6, 0-6, 6-3, 7-5 by Marcel Granollers of Spain.

It also means no U.S. man reached the fourth round at any of the four Grand Slam tournaments in 2013.

Granollers, who is ranked 43rd, now takes on No. 1 Novak Djokovic, a 6-0, 6-2, 6-2 winner over 95th-ranked Joao Sousa of Portugal.

The man Djokovic lost to in last year's U.S. Open final, and this year's Wimbledon final, Andy Murray, struggled with his breathing on a muggy afternoon but otherwise faced little trouble in a 7-6 (2), 6-2, 6-2 victory over 47th-ranked Florian Mayer of Germany. Murray has won 29 of his last 31 Grand Slam matches, a run of success that includes his first two major titles  at the U.S. Open last September and Wimbledon this July  along with two runner-up finishes.

"The expectations are higher, but there's not as much pressure to win," the third-seeded Murray said. "I feel much more comfortable coming into these events than this time last year."

"It was a bit hot, a bit humid, but otherwise, tennis was good," Berdych said after his 6-0, 6-3, 6-2 victory over No. 31 Julien Benneteau. "I managed to stay focused from the first point 'til the last. Didn't do any ups and downs."

The same almost could have been said by Williams, whose match against Stephens was eagerly anticipated from the moment the draw came out 1½ weeks ago. At the outset, anyway, it lived up to the hype.

Williams was the oldest player to reach the fourth round; Stephens the youngest. Williams owns 16 Grand Slam titles, including four at the U.S. Open, and 54 trophies total. Despite being the only current member of the top 20 in the WTA rankings without an appearance in a tournament final, Stephens has earned the label of "Next Big Thing," in large part by being one of three women to get to round of 16 at all four Grand Slam tournaments in 2013 (Williams, of course, is another).

Stephens' highlight to date was a semifinal run at the Australian Open in January, when she stunned Williams in the quarterfinals. Only adding to Sunday's story line, Stephens has said some unflattering things about Williams.

As they sat in their sideline chairs after warming up under a ceiling of gray clouds at Arthur Ashe Stadium, spectators began choosing sides. One yelled, "Come on, Serena!" Another, "Go, Sloane!" Williams opened Game 1 with a 96 mph ace and closed it with a 112 mph service winner. In between, Stephens managed to return a 118 mph serve. In Game 2, Stephens cranked a forehand winner down the line that drew gasps from the stands, and moments later, won a 24-stroke exchange thanks to fantastic retrieving.

Game 4 alone lasted 18 points over 11 minutes, and featured a 119 mph service winner by Stephens, who also delivered a cross-court backhand winner to cap a 10-shot point. When that game ended, it wasn't yet time to change ends, but both women wandered over to the sideline to towel off.

Soon, Williams nosed ahead, breaking to 4-2 with a cross-court forehand return winner off a 101 mph serve, the clenching a fist and shouting, "Come on!" But she handed a break right back in the next game by double-faulting on each of the last two points.

"I have to stop that," Williams said.

After she held to 5-4, everything changed, and Stephens was mostly her own undoing. One point from making it 5-all, she rushed an easy forehand, pushing it long, and slumped her head and shoulders. Then she sailed another forehand long. And, finally, she sent a forehand wide. Each point lasted three strokes: serve, return, bad groundstroke. Just like that, Williams broke to take the first set.

Those were three of Stephens' 29 unforced errors, 16 more than Williams made.

Stephens made one last stand in the opening game of the second set, earning a break point. But Williams wound up holding with the help of a 125 mph service winner and a 123 mph ace. Stephens wouldn't get another break chance. Indeed, take away the one service game Williams lost  the one with the duo of double-faults  and she won 35 of 44 points she served.

Afterward, Stephens spoke about embracing others' expectations and her personal goal of moving into the top 10 in the rankings by year's end. She also spoke about Williams, of course, and kept returning to a concept that was rather clear on this day, saying more than once: "She's No. 1 in the world for a reason." 